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Difference between revisions of "Alexander-Conway polynomial"

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The normalized version of the Alexander polynomial (cf. also [[Alexander invariants|Alexander invariants]]). It satisfies the Conway skein relation (cf. also [[Conway skein triple|Conway skein triple]])
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The normalized version of the Alexander polynomial (cf. also [[Alexander invariants]]). It satisfies the Conway skein relation (cf. also [[Conway skein triple]])
  
 
$$\Delta_{L_+}-\Delta_{L_-}=z\Delta_{L_0}$$
 
$$\Delta_{L_+}-\Delta_{L_-}=z\Delta_{L_0}$$
  
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and the initial condition $\Delta_{T_1}=1$, where $T_1$ is the trivial knot (cf. also [[Knot theory|Knot theory]]). For $z=\sqrt t-1/\sqrt t$ one gets the original Alexander polynomial (defined only up to $\pm t^i$).
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and the initial condition $\Delta_{T_1}=1$, where $T_1$ is the trivial knot (cf. also [[Knot theory]). For $z=\sqrt t-1/\sqrt t$ one gets the original Alexander polynomial (defined only up to $\pm t^i$).
  
 
====References====
 
====References====
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.W. Alexander,  "Topological invariants of knots and links"  ''Trans. Amer. Math. Soc.'' , '''30'''  (1928)  pp. 275–306</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.H. Conway,  "An enumeration of knots and links"  J. Leech (ed.) , ''Computational problems in abstract algebra'' , Pergamon  (1969)  pp. 329–358</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  L.H. Kauffman,  "The Conway polynomial"  ''Topology'' , '''20''' :  1  (1981)  pp. 101–108</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  J.W. Alexander,  "Topological invariants of knots and links"  ''Trans. Amer. Math. Soc.'' , '''30'''  (1928)  pp. 275–306</TD></TR>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top">  J.H. Conway,  "An enumeration of knots and links"  J. Leech (ed.) , ''Computational problems in abstract algebra'' , Pergamon  (1969)  pp. 329–358</TD></TR>
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<TR><TD valign="top">[a3]</TD> <TD valign="top">  L.H. Kauffman,  "The Conway polynomial"  ''Topology'' , '''20''' :  1  (1981)  pp. 101–108</TD></TR>
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</table>
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[[Category:Algebraic topology]]

Revision as of 20:23, 16 October 2014

The normalized version of the Alexander polynomial (cf. also Alexander invariants). It satisfies the Conway skein relation (cf. also Conway skein triple)

$$\Delta_{L_+}-\Delta_{L_-}=z\Delta_{L_0}$$

and the initial condition $\Delta_{T_1}=1$, where $T_1$ is the trivial knot (cf. also [[Knot theory]). For $z=\sqrt t-1/\sqrt t$ one gets the original Alexander polynomial (defined only up to $\pm t^i$).

References

[a1] J.W. Alexander, "Topological invariants of knots and links" Trans. Amer. Math. Soc. , 30 (1928) pp. 275–306
[a2] J.H. Conway, "An enumeration of knots and links" J. Leech (ed.) , Computational problems in abstract algebra , Pergamon (1969) pp. 329–358
[a3] L.H. Kauffman, "The Conway polynomial" Topology , 20 : 1 (1981) pp. 101–108
How to Cite This Entry:
Alexander-Conway polynomial. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Alexander-Conway_polynomial&oldid=32689
This article was adapted from an original article by Jozef Przytycki (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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